# A Neural Operator-Based Approach to Symbolic Discovery of PDEs

arXiv: https://arxiv.org/abs/2501.08086 · HTML: https://mlsanity.com/paper/2501.08086/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 16 references that are in the map.
- First posted on arXiv 14 Jan 2025.

## Builds on (most cited first, 10 of 10)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Multipole Graph Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2006.09535/) (2020)
- [Convolutional Neural Operators for robust and accurate learning of PDEs](https://mlsanity.com/paper/2302.01178/) (2023)
- [PDE-Net 2.0: Learning PDEs from Data with A Numeric-Symbolic Hybrid Deep Network](https://mlsanity.com/paper/1812.04426/) (2018)
- [Physics-informed learning of governing equations from scarce data](https://mlsanity.com/paper/2005.03448/) (2020)
- [Learning Equations for Extrapolation and Control](https://mlsanity.com/paper/1806.07259/) (2018)
- [DeepMoD: Deep learning for Model Discovery in noisy data](https://mlsanity.com/paper/1904.09406/) (2019)
- [Deep Learning and Symbolic Regression for Discovering Parametric Equations](https://mlsanity.com/paper/2207.00529/) (2022)
- [Neural Integral Equations](https://mlsanity.com/paper/2209.15190/) (2022)

## Built on by (most cited first, 2 of 2)

- [Data-driven Discovery of Digital Twins in Biomedical Research](https://mlsanity.com/paper/2508.21484/) (2025)
- [Neuro-Symbolic ODE Discovery with Latent Grammar Flow](https://mlsanity.com/paper/2604.16232/) (2026)

## Abstract (arXiv)

Discovering governing equations from data remains challenging when the underlying dynamics involve nonlocal differential operators, field interactions governed by auxiliary equations, or temporal memory effects. We propose Neural Operator-based symbolic Model approximaTion and discOvery (NOMTO), a framework that extends Equation Learner-type symbolic architectures by incorporating pretrained neural operators as nodes in the symbolic network. NOMTO represents candidate equations as sparse differentiable computational graphs that combine algebraic operations with fixed neural operator surrogates pretrained to approximate nonlinear operators. We evaluate the method on model-discovery problems involving nonlocal spatial operators, couplings mediated by auxiliary field equations, and temporal integral terms representing memory effects. The results show that NOMTO can recover compact governing equations containing nonlocal operator terms, thereby extending symbolic model discovery beyond libraries restricted to local derivatives and point-wise algebraic combinations.